Large spin systematics in CFT

The analytic functional bootstrap. Part II. Natural bases for the crossing equation

“…The basic idea is that once we have determined all double-trace data at a given order in perturbation theory, we can use the functionals α n and β n to compute the OPE coefficient and anomalous dimension of [φ 2 ] n at the next order. Let us write the perturbative expansion of the four-point function up to two loops as follows G(z) = G (0) (z) + G (1) (z)g 2 + G (2) (z)g 4 + G (3) (z)g 6 + O(g 8 ) .…”

“…As a result, existing analytic approaches typically rely on expanding the equations around a specific point in the space of conformal cross-ratios. Indeed, the subject of modern analytic conformal bootstrap started by studying the double light-cone limit [1][2][3][4][5]. More recently, progress has been made also by utilizing the Regge limit [6][7][8][9] and the deep Euclidean limit [10,11].…”

“…We are led to the conclusion that both the light-cone and Regge limits should play an important role in the numerical bounds on the low-lying CFT spectrum. 3 We can think of the expansion around the crossing-symmetric point as providing a specific basis for the space dual to the space of functions holomorphic in the crossing region. Clearly, it would be of great interest to have instead a basis which extracts the complete in-formation contained in the crossing region, in a manner reflecting our analytic understanding of the corners.…”

“…This assumption is presumably possible to prove from the fact that all interaction vertices are relevant, and is ultimately justified by the self-consistency of the answers we find. Let us write the perturbative expansion of the double-trace data up to two loops as follows ∆ n (g) = 2∆ φ + 2n + γ (1) n g 2 + γ (2) n g 4 + γ (3) n g 6 + O(g 8 ) a n (g) = a (0) n + a (1) n g 2 + a (2) n g 4 + a (3) n g 6 + O(g 8 ) .…”

“…a (2) n G ∆n (z) + a (0) n γ (2) n + a (1) n γ (1) n ∂G ∆n (z) + 1 2 a (0) n (γ (1) n ) 2 ∂ 2 G ∆n (z) (7.16) 3) n G ∆n (z) + a (0) n γ (3) n + a (1) n γ (2) n + a (2) n γ (1) n ∂G ∆n (z) + (7.17) + a (0) n γ (1) n γ (2) n + 1 2 a (1) n (γ (1)…”